cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A371708 Expansion of g.f. A(x) satisfying A( x*A(x - x^2) ) = x^2.

Original entry on oeis.org

1, 1, 1, 2, 6, 19, 60, 193, 636, 2141, 7331, 25451, 89385, 317036, 1134100, 4087104, 14825482, 54088470, 198348985, 730723956, 2703194553, 10037648254, 37399878530, 139785998185, 523962161491, 1969154471389, 7418488063284, 28010998254007, 105986233046356, 401804972780552
Offset: 1

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Author

Paul D. Hanna, Apr 23 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 60*x^7 + 193*x^8 + 636*x^9 + 2141*x^10 + 7331*x^11 + 25451*x^12 + 89385*x^13 + 317036*x^14 + ...
where A( x*A(x - x^2) ) = x^2.
RELATED SERIES.
Let R(x) be the series reversion of A(x), A(R(x)) = x, which begins
R(x) = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 5*x^6 + 6*x^7 - 16*x^8 + 23*x^9 - 62*x^10 + 100*x^11 - 270*x^12 + 463*x^13 - 1254*x^14 + 2224*x^15 - 6050*x^16 + ...
then R( R(x^2)/x ) = x - x^2.
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x + x^3 + 2*x^5 + 6*x^7 + 23*x^9 + 100*x^11 + 463*x^13 + 2224*x^15 + 10963*x^17 + ...
B2 = (R(x) + R(-x))/2 = -x^2 - 2*x^4 - 5*x^6 - 16*x^8 - 62*x^10 - 270*x^12 - 1254*x^14 - 6050*x^16 + ...
and satisfy B1^2 + B2 = 0 and A(x*B1) = B1^2.
SPECIFIC VALUES.
A( A(1/4) / 2 ) = 1/4 where
A(1/4) = 0.39241307250698647662923990494867613212061604622566765...
A( A(2/9) / 3 ) = 1/9 where
A(2/9) = 0.29957319341272312632777466712131772539171747971866175...
A( A(3/16) / 4 ) = 1/16 where
A(3/16) = 0.2352360051274118086289466324430753987734355106832392...
A( A(4/25) / 5 ) = 1/25 where
A(4/25) = 0.1922953260179964363449115205476634347705922222443464...

Links

Crossrefs

Cf. A372528, A272483, A213591, A000108.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( x^2 - subst(Ser(A),x, x*subst(Ser(A),x, x - x^2) ), #A));A[n+1]}
for(n=1,35,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.
(1) A( x*A(x - x^2) ) = x^2.
(2) A(x - x^2) = R(x^2)/x.
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( A(x)*C(x) ) = C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
a(n) ~ c * 4^n / n^(3/2), where c = 0.0517683007874758928168667... - Vaclav Kotesovec, Apr 24 2024

A272484 G.f. A(x) satisfies: A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 86, 278, 928, 3164, 10958, 38428, 136168, 486796, 1753660, 6359961, 23202408, 85093552, 313548346, 1160248084, 4309812532, 16064728072, 60070599076, 225271863550, 847042748378, 3192758928650, 12061704111576, 45662648135238, 173204482763760, 658180582310888, 2505341336035650, 9551632787000829, 36469897605758744, 139443687986144472, 533869533407865024, 2046496258409861740, 7854102611559917914
Offset: 1

Views

Author

Paul D. Hanna, May 05 2016

Keywords

Comments

The radius of convergence of g.f. A(x) is 1/4.
Specific value S = A(1/4) = 0.44982760488955294204795759797171897522321034552221... satisfies:
(1) S^2 = 2 * A(S^3),
(2) S^4 = 8 * A(S^6/8) / (1 - sqrt(1 - 4*S^3)).
Limit a(n)/A000108(n-1) appears to be near 0.6564...
The numerical value of this limit is 0.6564415409950121... . - Vaclav Kotesovec, May 07 2016

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 28*x^6 + 86*x^7 + 278*x^8 + 928*x^9 + 3164*x^10 + 10958*x^11 + 38428*x^12 +...
such that A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2.
RELATED SERIES.
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 +...+ A000108(n-1)*x^n +...
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 32*x^6 + 92*x^7 + 284*x^8 + 920*x^9 + 3080*x^10 + 10544*x^11 + 36684*x^12 + 129228*x^13 + 459860*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 72*x^7 + 216*x^8 + 680*x^9 + 2226*x^10 + 7506*x^11 + 25858*x^12 + 90498*x^13 + 320580*x^14 + 1146670*x^15 +...
A( A(x)^3 ) = x^3 + 3*x^4 + 9*x^5 + 26*x^6 + 78*x^7 + 243*x^8 + 786*x^9 + 2619*x^10 + 8928*x^11 + 30967*x^12 + 108870*x^13 + 386928*x^14 + 1387560*x^15 +...
where A( A(x)^3 ) = C(x)*A(x)^2.
A(x-x^2) = x - x^4 + 2*x^7 - 4*x^10 + 4*x^13 + 23*x^16 - 212*x^19 + 1148*x^22 - 4906*x^25 + 16904*x^28 - 41046*x^31 + 6730*x^34 + 713246*x^37 - 5703472*x^40 +...
where A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
A( A(x-x^2)^3 ) = x^3 - 2*x^6 + 5*x^9 - 12*x^12 + 20*x^15 + 22*x^18 - 438*x^21 + 2780*x^24 - 13124*x^27 + 50092*x^30 - 145875*x^33 + 201848*x^36 +...
where A( A(x-x^2)^3 ) = x * A(x-x^2)^2.

Links

Crossrefs

Cf. A272483.

Programs

  • PARI
    {a(n) = my(A=x, C=x, X=x+x*O(x^n)); for(i=1, n, C = X + C^2; A = (2*A - subst(A, x, A^3)/(C*A) )); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A( A(x-x^2)^3 ) = x * A(x-x^2)^2.
(2) A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
(3) A( A(x^3)/x^2 - A(x^3)^2/x^4 ) = x.
a(n) ~ c * 4^n / n^(3/2), where c = 0.09258936990935582... . - Vaclav Kotesovec, May 07 2016
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